W1D1: Jan. 4th, 2022

About Class

  • PHd in Statistics from Idaho

  • Homework Due on Friday, and will be graded over the weekend.

  • Starting with Chapter 5.

  • Class notes are in Canvas

461 Last Lecture Notes

Joint CDF and PDF

CDF for two random variables have 3 properties

  • smallest value is 0

  • largest value is 1

  • always non-decreasing

Joint PMF for discrete variables uses summation and has two properties

  • all probability are between 0 and 1

  • total probability must be 1

Example: Two-way Table

Two way tables, because there are two ways to read it

  • also known as pivot table and contingency table

  • make sure you can write and understand in different ways

a) P(Only 1 car will be sold)

b) P(1 warrenty will be sold given that 2 cars were sold on a given day)

c) Find and report the marginal pmf of \(Y_1\)

“O otherwise” must be included for a complete distribution function.

d) Are \(Y_1\) and \(Y_2\) independent?

W1D2: Jan. 6th, 2022

Review Homework will be Homework 1 and due Friday 14th

Joint PDF

Properties :

  1. smallest value is 0

  2. total volume under the joint pdf is 1

Notes :

If the limits are given without equal sign then you can apply integral in the same way.

Example : Joint pdf

a) Draw and shade the support

The support is the area on the graph.

Upper limit is only important for PMF of discrete random variables

  • pdf : \(f_{Y_1,Y_2}\geq0\)

  • PMF : \(0\leq p(y_1,y_2)\leq 1\)

b) Verify \(f(y_1,y_2)\) is a valid joint pdf

c) Find the joint CDF

CDF : \(F(y_1,y_2)=P(Y_1\leq y_1, Y_2\leq y_2)=\int_{-\infty}^{y_2}\int_{-\infty}^{y_1}f(y_1,y_2)dt_1dt_2\)

  • when we do problems we do not need to write every step of the integral

  • pay attention to the limits on the double integral (\(y_1\) is on the inner integral, meaning it is integrated first (\(dt_1\) inside \(dt_2\)))

  • final answer will be a function, not a limit

  • lowest value must be 0, and highest value must be 1

    • 0 otherwise is wrong without values for 1 included in CDF
  • when you find the cdf you must use \(dt_1\) and \(dt_2\), but don’t need it to use or solve for probabilities

d) P(Both variables are less than 1/4)

e) P(\(Y_1\) is more than 0.75)

W2D3: Jan. 11th, 2022

W2D4: Jan. 13th, 2022

W3D5: Jan. 18th, 2022

joint pmf (p) : between 0 and 1

joint pdf (f) : nonnegative

Discrete uses summation

W3D6: Jan. 20th, 2022

W4D7: Jan. 25th, 2022

W4D8: Jan. 27th, 2022

W5D9: Feb. 1st, 2022

MIDTERM